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Theorem sxsigon 31508
Description: A product sigma-algebra is a sigma-algebra on the product of the bases. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Assertion
Ref Expression
sxsigon ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)))

Proof of Theorem sxsigon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sxsiga 31507 . 2 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
2 eqid 2824 . . . . . 6 ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
32sxval 31506 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
43unieqd 4838 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
5 mpoexga 7771 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
6 rnexg 7609 . . . . 5 ((𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
7 unisg 31459 . . . . 5 (ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V → (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
85, 6, 73syl 18 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
94, 8eqtrd 2859 . . 3 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
10 eqid 2824 . . . 4 𝑆 = 𝑆
11 eqid 2824 . . . 4 𝑇 = 𝑇
122, 10, 11txuni2 22173 . . 3 ( 𝑆 × 𝑇) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
139, 12syl6reqr 2878 . 2 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
14 issgon 31439 . 2 ((𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)) ↔ ((𝑆 ×s 𝑇) ∈ ran sigAlgebra ∧ ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇)))
151, 13, 14sylanbrc 586 1 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480   cuni 4824   × cxp 5540  ran crn 5543  cfv 6343  (class class class)co 7149  cmpo 7151  sigAlgebracsiga 31424  sigaGencsigagen 31454   ×s csx 31504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-siga 31425  df-sigagen 31455  df-sx 31505
This theorem is referenced by:  sxuni  31509
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